Algorithmic Game Theory and Applications
Lecture 17:
A first look at Auctions and Mechanism Design:
Auctions as Games, Bayesian Games, Vickrey auctions
Kousha Etessami
Food for thought: sponsored search auctions
Question
How should advertisement slots on your Google search page be auctioned?
You may have already heard that Google uses a so-called
“Generalized Second Price Auction” mechanism to do this.
But why do they do so? Is there any “better” way?
What does “better” actually mean? What is the goal of the auctioneer?
How should other (electronic) auctions be conducted?
More generally, how should we design games when our goal is to compel
selfish players to behave in a desired way (e.g., a “socially optimal” way, or
“truthfully”). This is the topic of Mechanism Design.
Auction theory and Mechanism Design are rich and vast subdisciplines of
game/economic theory. We can not do them adequate justice. We briefly
explore them in remaining lectures, focusing on algorithmic aspects.
Some reference reading: Chapters 9, 11, 12, 13, & 28 of AGT book.
Kousha Etessami ()
Lecture 17
2 / 10
Auctions as games
Consider one formulation of a single-item, sealed-bid, auction as a game:
Each of n bidders is a player.
Each player i has a valuation, vi ∈ R, for the item being auctioned.
if the outcome is: player i wins the item and pays price pr , then the
payoff to player i, is
ui (outcome) := vi − pr
and all other players j 6= i get payoff 0: uj (outcome) := 0.
Let us require that the auctioneer must set up the rules of the auction
so that they satisfy the following reasonable constraints: given
(sealed) bids (b1 , . . . , bn ), one of the highest bidders must win, and
must pay a price pr such that 0 ≤ pr ≤ maxi bi .
Question: What rule should the auctioneer employ, so that for each
player i, bidding their “true valuation” vi (i.e., letting bi := vi ) is a
dominant strategy?
Kousha Etessami ()
Lecture 17
3 / 10
Vickrey auctions
In a Vickrey auction, a.k.a., second-price, sealed bid auction, a
highest bidder, j, whose bid is bj = maxi bi , gets the item, but pays
the second highest bid price: pr = maxi6=j bi .
Claim
Bidding their true valuation, vi , i.e., letting bi := vi , is a (weakly)
dominant strategy in this game for all players i.
Let us prove this on the board. (We actually prove something stronger.)
Note: there is something very fishy/unsatisfactory about our
formulation so far of an auction as a complete information game:
player i normally does not know the valuation vj of other players j 6= i.
But if viewed as a complete information game, then every player
knows every one else’s valuation. This is totally unrealistic!
We need a better game-theoretic model for settings like auctions.
Kousha Etessami ()
Lecture 17
4 / 10
Bayesian Games (Games of Incomplete Information) [Harsanyi,’67,’68]
A Bayesian Game, G = (N, (Ai )i∈N , (Ti )i∈N , (ui )i∈N , p), has:
A
A
A
A
(finite) set N = {1, . . . , n} of players.
(finite1 ) set Ai of actions for each player i ∈ N.
(finite1 ) set of possible types, Ti , for each player i ∈ N.
payoff (utility) function, for each player i ∈ N:
ui : A1 × . . . × An × T1 × . . . × Tn → R
A (joint) probability distribution over types:
p : T1 × . . . × Tn → [0, 1]
where, letting T = T1 × . . . × Tn , we must have:
X
p(t1 , . . . , tn ) = 1
(t1 ,...,tn )∈T
p is sometimes called a common prior.
1
We can and do often remove the finiteness assumption on the type/action spaces,
e.g., letting Ti be a closed interval [a, b] ⊆ R.
Kousha Etessami ()
Lecture 17
5 / 10
strategies and expected payoffs in Bayesian games
A pure strategy for player i is a function si : Ti → Ai .
I.e., player i knows its own type, ti , and chooses action si (ti ) ∈ Ai .
(In a mixed strategy, xi , xi (ti ) is a probability distribution over Ai .)
Players’ types are chosen randomly according to (joint) distribution p.
Player i knows ti ∈ Ti , but doesn’t know the type tj of players j 6= i.
But every player “knows” the joint distribution p (this is often too
strong an assumption, dubiously so, but it is useful), so each player i
can compute the conditional probabilities, p(t−i | ti ), on other
player’s types, given its own type ti .
The expected payoff to player i, under the pure profile
s = (s1 , . . . , sn ), when player i has type ti (which it knows) is:
Ui (s, ti ) =
X
p(t−i | ti )ui (s1 (t1 ), . . . , sn (tn ), ti , t−i )
t−i
Kousha Etessami ()
Lecture 17
6 / 10
Bayesian Nash Equilibrium
Definition
A strategy profile s = (s1 , . . . , sn ) is a (pure) Bayesian Nash equilibrium
(BNE) if for all players i and all types ti ∈ Ti , and all strategies si0 for
player i, we have: Ui (s, ti ) ≥ Ui ((si0 ; s−i ), ti ).
(A mixed BNE is defined similarly, by allowing players to use mixed
strategies, xi , and defining expected payoffs Ui (x, ti ) accordingly.)
Proposition
Every finite Bayesian Game has a mixed strategy BNE.
Proof
Follows from Nash’s Theorem. Every finite Bayesian Games can be
encoded as a finite extensive form game of imperfect information: the
game tree first randomly choose a subgame labeled (t1 , . . . , tn ), with
probability p(t1 , . . . , tn ). All nodes belonging to player i in different
subgames labeled by the same type ti ∈ Ti are in the same information set.
Kousha Etessami ()
Lecture 17
7 / 10
Back to Vickrey auctions
Now suppose we model a sealed-bid single-item auction using a Bayesian
game, with some arbitrary prior probability distribution p(v1 , . . . , vn ) over
valuations (suppose every vi is in some finite nonnegative range [0, vmax ]).
The private information of each player i is ti := vi .
Proposition
In the Vickrey (second-price, sealed-bid) auction game, with any prior p,
the truth revealing profile of bids v = (v1 , . . . , vn ), is a weakly dominant
strategy profile.
(In fact, under suitable conditions on p, v is the unique BNE of the game.)
Proof
The exact same proof as for the complete information version of the
vickrey game works to show that v is a weakly dominant strategy profile.
Indeed, we didn’t use the fact that the game has complete information.
We didn’t even assume the players have a common prior for this!
Kousha Etessami ()
Lecture 17
8 / 10
What about other auctions?
In first-price, sealed-bid auctions, (maximum bidder gets the item and
pays pr = maxi bi ), bidding truthfully may indeed not be a dominant
strategy. E.g., if player i knows (with high probability) that its
valuation vi >> vj for all j 6= i, then i will not “waste money” and
will instead bid bi << vi , since it will still get the item.
What implications does this have for the expected revenue of the
auction? Surprisingly, it has less than you might think:
Proposition
If prior p is a product of i.i.d. uniform distributions over some interval
[0, vmax ], then the expected revenue of the second-price and first-price
sealed-bid auctions are both the same in their (unique) symmetric BNEs.
This is actually a special case of a much more general result in Mechanism
Design called the Revenue Equivalence Principle. (But anyway, note that
one-shot revenue maximization is not always a wise goal for an auctioneer.)
Kousha Etessami ()
Lecture 17
9 / 10
The bigger picture
This was our first look at Auctions (with hints of Mechanism design),
in the simple setting of a single-item, sealed-bid, auction.
To delve deeper into (algorithmic aspects of) Mechanism Design and
auctions, we need a deeper understanding of the “bigger picture”:
Starting next time, we will step back to glimpse at where all this fits
in the broader scope of Economic Theory.
In particular, we will briefly discuss some social choice theory, and also
Market equilibria, before returning to the important VCG mechanism,
which vastly generalizes the Vickrey single-item auction.
Some of the topics we will discuss are beyond the scope of this AGT
course, and can be properly learned, e.g., in a good Microeconomic
Theory course/text (e.g., [Mas-Colell-Whinston-Green’95]).
However, I feel I would be “cheating you” if I didn’t at least provide
some (necessarily impressionistic) sense of the “bigger picture”.
Kousha Etessami ()
Lecture 17
10 / 10